Some Topics On The Ramsey Number Of Tree And Cycles

Ramsay theory is an important part of combinatorics,and it is widely used in theoretical computer science,information theory,science of decision making,financial economics and other fields.Ramsay numbers quantify the existence theorem in Ramsay theory,Solving the exact value of the Ramsey number of graphs and improving its approximate bounds are active branches of Ramsey theory.At the same time,the study of Ramsey numbers of graphs also plays an important role in computational geometry,logical analysis,parallel computing,complex structures and solving other NP-hard problems.The key to solving the Ramsey number problem is to calculate the exact value by finding an effective research tool.Christou et al.show that for any 2-coloring in a complete bipartite graph,there are bipartite Ramsey numbers with monochromatic matchings.Lortz et al.give some relevant results for size Ramsey numbers.Thomason et al.studied the Gallai-Ramsey numbers for G=Kn(n?5)that do not contain rainbow P5 or rainbow K1,3 after satisfying edge coloring.On this basis,this thesis mainly uses the pigeonhole principle and the proof by countradiction to give some relevant results of Ramsey numbers:(1)In calculating the bipartite Ramsey number,for any k-coloring in a complete bipartite graph,there is a monochromatic nPl,which gives the exact value of the bipartite Ramsey number when the l is smaller and the approximate bounds of the bipartite Ramsey number when the l is larger;(2)When calculating the size Ramsey number,the arbitrary graph is divided by using the maximum degree of the star graph,and the exact value of r(P2?P3,K1,m?K1,n)is given.The definition of(n,t)-brush graph takes into account the exact value of r(aP2?bP3,K1,m)and the approximate bounds of r(mP2?P4,K1,m),r(mP2?P5,K1,m);(3)When calculating the Gallai-Ramsey number,some related theorems such as Gallai partition used to prove that the Gallai-Ramsey numbers contains path and cycle graph when the G=Kn(n?5)satisfies the edge coloring without rainbow P5 or rainbow K1,3.Some results of classical Ramsey numbers also help to calculate the Gallai-Ramsey numbers.